New Lower Bounds for the Least Common Multiples of Arithmetic Progressions

Abstract

For relatively prime positive integers $u_0$ and $r$ and for $0\le k\le n$, define $u_k:=u_0+kr$. Let $L_n:={\rm lcm}(u_0, u_1, ..., u_n)$ and let $a, l\ge 2$ be any integers. In this paper, we show that, for integers $\alpha \geq a$ and $r\geq \max(a, l-1)$ and $n\geq l\alpha r$, we have $$L_n\geq u_0r^{(l-1)\alpha +a-l}(r+1)^n.$$ Particularly, letting $l=3$ yields an improvement to the best previous lower bound on $L_n$ obtained by Hong and Kominers.